Question: Simplify; express your answer in exponential form. Assume $t\neq 0, k\neq 0$. $\dfrac{{(t^{-3}k^{-5})^{-3}}}{{(t^{4}k^{-2})^{-1}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{-3}k^{-5})^{-3} = (t^{-3})^{-3}(k^{-5})^{-3}}$ On the left, we have ${t^{-3}}$ to the exponent ${-3}$ . Now ${-3 \times -3 = 9}$ , so ${(t^{-3})^{-3} = t^{9}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{-3}k^{-5})^{-3}}}{{(t^{4}k^{-2})^{-1}}} = \dfrac{{t^{9}k^{15}}}{{t^{-4}k^{2}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{9}k^{15}}}{{t^{-4}k^{2}}} = \dfrac{{t^{9}}}{{t^{-4}}} \cdot \dfrac{{k^{15}}}{{k^{2}}} = t^{{9} - {(-4)}} \cdot k^{{15} - {2}} = t^{13}k^{13}$